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Metamath Proof Explorer

Theorem ralnex

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997) (Proof shortened by BJ, 16-Jul-2021)

		Ref	Expression
	Assertion	ralnex	$⊢ \forall x \in A \neg φ \leftrightarrow \neg \exists x \in A φ$

Proof

Step	Hyp	Ref	Expression
1		raln	$⊢ \forall x \in A \neg φ \leftrightarrow \forall x \neg (x \in A \land φ)$
2		alnex	$⊢ \forall x \neg (x \in A \land φ) \leftrightarrow \neg \exists x (x \in A \land φ)$
3		df-rex	$⊢ \exists x \in A φ \leftrightarrow \exists x (x \in A \land φ)$
4	2 3	xchbinxr	$⊢ \forall x \neg (x \in A \land φ) \leftrightarrow \neg \exists x \in A φ$
5	1 4	bitri	$⊢ \forall x \in A \neg φ \leftrightarrow \neg \exists x \in A φ$